Polar codes are based on Kronecker product matrices. G=F⊗m=F ⊗ . . . ⊗F is the m-fold Kronecker product of a seed matrix F.
If A is an m×n matrix and B is a p×q matrix, then the Kronecker product A⊗B is the mp×nq block matrix:
            A      ⊗      B        =          [                                                                  a                11                            ⁢              B                                            …                                                              a                                  1                  ⁢                  n                                            ⁢              B                                                            ⋮                                ⋱                                ⋮                                                                              a                                  m                  ⁢                                                                          ⁢                  1                                            ⁢              B                                            …                                                              a                mn                            ⁢              B                                          ]        ,
more explicitly:
      A    ⊗    B    =            [                                                                  a                11                            ⁢                              b                11                                                                                        a                11                            ⁢                              b                12                                                          …                                                              a                11                            ⁢                              b                                  1                  ⁢                  q                                                                          …                                …                                                              a                                  1                  ⁢                  n                                            ⁢                              b                11                                                                                        a                                  1                  ⁢                  n                                            ⁢                              b                12                                                          …                                                              a                                  1                  ⁢                  n                                            ⁢                              b                                  1                  ⁢                  q                                                                                                        a                              11                ⁢                                  b                  21                                                                                                        a                11                            ⁢                              b                22                                                          …                                                              a                11                            ⁢                              b                                  2                  ⁢                  q                                                                          …                                …                                                              a                                  1                  ⁢                  n                                            ⁢                              b                21                                                                                        a                                  1                  ⁢                  n                                            ⁢                              b                22                                                          …                                                              a                                  1                  ⁢                  n                                            ⁢                              b                                  2                  ⁢                  q                                                                                          ⋮                                ⋮                                ⋱                                ⋮                                                                                                                                                    ⋮                                ⋮                                ⋱                                ⋮                                                                              a                11                            ⁢                              b                                  p                  ⁢                                                                          ⁢                  1                                                                                                        a                11                            ⁢                              b                                  p                  ⁢                                                                          ⁢                  2                                                                          …                                                              a                11                            ⁢                              b                pq                                                          …                                …                                                              a                                  1                  ⁢                  n                                            ⁢                              b                                  p                  ⁢                                                                          ⁢                  1                                                                                                        a                                  1                  ⁢                  n                                            ⁢                              b                                  p                  ⁢                                                                          ⁢                  2                                                                          …                                                              a                                  1                  ⁢                  n                                            ⁢                              b                pq                                                                          ⋮                                ⋮                                                                                          ⋮                                ⋱                                                                                          ⋮                                ⋮                                                                                          ⋮                                                ⋮                                ⋮                                                                                          ⋮                                                                                          ⋱                                ⋮                                ⋮                                                                                          ⋮                                                                              a                                  m                  ⁢                                                                          ⁢                  1                                            ⁢                              b                11                                                                                        a                                  m                  ⁢                                                                          ⁢                  1                                            ⁢                              b                12                                                          …                                                              a                                  m                  ⁢                                                                          ⁢                  1                                            ⁢                              b                                  1                  ⁢                  q                                                                          …                                …                                                              a                mn                            ⁢                              b                11                                                                                        a                mn                            ⁢                              b                12                                                          …                                                              a                mn                            ⁢                              b                                  1                  ⁢                  q                                                                                                                        a                                  m                  ⁢                                                                          ⁢                  1                                            ⁢                              b                21                                                                                        a                                  m                  ⁢                                                                          ⁢                  1                                            ⁢                              b                22                                                          …                                                              a                                  m                  ⁢                                                                          ⁢                  1                                            ⁢                              b                                  2                  ⁢                  q                                                                          …                                …                                                              a                mn                            ⁢                              b                21                                                                                        a                mn                            ⁢                              b                22                                                          …                                                              a                mn                            ⁢                              b                                  2                  ⁢                  q                                                                                          ⋮                                ⋮                                ⋱                                ⋮                                                                                                                                                    ⋮                                ⋮                                ⋱                                ⋮                                                                              a                                  m                  ⁢                                                                          ⁢                  1                                            ⁢                              b                                  p                  ⁢                                                                          ⁢                  1                                                                                                        a                                  m                  ⁢                                                                          ⁢                  1                                            ⁢                              b                                  p                  ⁢                                                                          ⁢                  2                                                                          …                                                              a                                  m                  ⁢                                                                          ⁢                  1                                            ⁢                              b                pq                                                          …                                …                                                              a                mn                            ⁢                              b                                  p                  ⁢                                                                          ⁢                  1                                                                                                        a                mn                            ⁢                              b                                  p                  ⁢                                                                          ⁢                  2                                                                          …                                                              a                mn                            ⁢                              b                pq                                                        ]        .  
FIG. 1 shows how a Kronecker product matrix can be produced from a seed matrix G2 102. Shown in FIG. 1 are the 2-fold Kronecker product matrix G2 ⊗2 102 and the 3-fold Kronecker product matrix G2 ⊗3 104, where
      G    2    =            (                                    1                                1                                                0                                1                              )        .  This approach can be extended to produce m-fold Kronecker product matrix G2 ⊗m.
A polar code can be formed from a Kronecker product matrix based on matrix G2. For a polar code having codewords of length N=2m, the generator matrix is G2 ⊗m. An example of using Kronecker product matrix G2 ⊗3 to produce codewords of length 8 is depicted in FIG. 2. A codeword x is formed by the product of an input vector u and the Kronecker product matrix G2 ⊗3 104 as indicated at 110. The input vector u is composed of frozen bits and information bits. In the specific example, N=8, so the input vector u is an 8 bit vector, and the codeword x is an 8-bit vector. The input vector has frozen bits in positions 0, 1, 2 and 4, and has information bits at positions 3, 5, 6 and 7. An example implementation of a coder that generates codewords is indicated at 112, where the frozen bits are all set to 0, where a circle around a plus symbol indicates modulo 2 addition. For the example of FIG. 2, an N=8 bit input vector is formed from K=4 information bits and N−K=4 frozen bits. Codes of this form are referred to as Polar codes and the encoder is referred to as a Polar encoder.
More generally, in “Channel Polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels” by E. Arikan, IEEE Transactions on Information Theory Volume 55, Issue 7, Published July 2009, a “channel polarization” theory was proved in chapter IV. Channel polarization is an operation which produces N channels from N independent copies of a binary-input discrete memoryless channel (B-DMC) W such that the new parallel channels are polarized in the sense that their mutual information is either close to 0 (low mutual SNR channels) or close to 1 (high mutual SNRchannels). In other words, some encoder bit positions will experience a channel with high mutual SNR, and will have a relatively low reliability/low possibility to be correctly decoded, and for some encoder bit positions, they will experience a channel with a high mutual SNR, and will have high reliability/high possibility to be correctly decoded. In code construction, information bits are put in the reliable positions and frozen bits (bits known to both encoder and decoder) are put in unreliable positions. In theory, the frozen bits can be set to any value so long as the frozen bit sequence is known to both encoder and decoder. In conventional applications, the frozen bits are all set to “0”.
In a communication scenario in which a transmitter may want to send the data only for one or several specific receivers, an encoded block transmitted over the air can be received by any other non-targeted receivers. Various approaches can be employed to prevent these non-targeted receivers from decoding the information. Some systems use a well-known security/encrypting protocol on higher layers to provide privacy. However, these approaches involve some higher-layer scheduling resources, and result in a long processing delay. Moreover, higher-layer scheduling algorithms may not provide sufficient security in a relatively low SNR condition.
It would be advantageous to have a relatively simple approach to providing security for polar codes.